Abstracts

Jing Wang (Purdue)
We study the sub-Laplacian L on a contact Riemannian manifold M. L is a symmetric diffusion operator which is subelliptic but nowhere elliptic. We obtain the Bakry-Émery type criterion (curvature-dimension inequality) for L which gives an analytic approach to the geometric property. As a consequence, we prove that under suitable geometric bounds, spectral gap estimates can be obtained as well as the convergence to the equilibrium of the associated Markov processes. This is a joint work with F. Baudoin.top of page

Fleming-Viot particle system driven by a random walk on naturals

Nevena Maric (Missouri)
Random walk on naturals with negative drift and absorption at 0, when conditioned on survival, has uncountably many invariant measures (quasi-stationary distributions, qsd). We study a Fleming-Viot (FV) particle system driven by this process. In this particle system there are N particles where each particle evolves as the random walk described above. As soon as one particle is absorbed, it reappears, choosing a new position according to the empirical measure at that time. Between the absorptions, the particles move independently of each other. Our focus is in the relation of empirical measure of the FV process with qsds of the random walk.

Firstly, mean normalized densities of the FV unique stationary measure converge to the minimal qsd, as N goes to infinity. Moreover, every other qsd of the random walk corresponds to a metastable state of the FV particle system.top of page

Yuri Bakhtin (GA Tech)
The classical Freidlin--Wentzell theory on small random perturbations of dynamical systems operates mainly at the level of large deviation estimates. It would be interesting and useful to supplement those with central limit theorem type results. We are able to describe a class of situations where a Gaussian scaling limit for the exit point of conditioned diffusions holds. Our main tools are Doob's h-transform and new gradient estimates for Hamilton--Jacobi equations. Joint work with Andrzej Swiech.top of page